20. Lurch 5 angular

The model and its justification
Lurch 5 Angular is the rotational analog of Lurch 5 Linear. Like Lurch 4 Angular, it calculates the speed of the rearward lurch from the bullet, the cloud, the large fragments, conical motion of cloud, and 3-D motion of the large fragments. To this it adds 3-D motion of the body. It turns out that the 3-D component of the body's KE does not change the answer perceptibly—the default velocity of -1.38 ft s^-1 in Lurch 4 Angular remains -1.38 ft s^-1 here. Thus the 3-D motion of the body is an insignificant term in the overall equation.

Solving the simultaneous equations
The three-dimensionality of the body's kinetic energy is expressed by the term f_kebody, which is similar in form and function to f_kefrag. We assume a range of 1–1.4 and a starting value of 1.2.

Conservation of angular X-momentum

Conservation of total energy

Solutions to simultaneous equations

Distributions of momentum and energy

Angular momentum		Energy, ft-lb
Before	After	Before	After
Ω_bullet = 3.80	Ω_bulletafter = 0.43	KE_bullet = 1164	KE_bulletafter = 14
	Ω_bodyafter = -4.05		KE_bodyafter = 1
	Ω_cloud = 6.28		KE_cloud = 668
	Ω_frag1 = 0.97		KE_frag1 = 132
	Ω_frags23 = 0.16		KE_frags23 = 49
			PE = 300

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
						0	-1.813
156	-1.356					50	-1.746	1750	-1.309
157	-1.361	65	-1.803			100	-1.677	1760	-1.323
158	-1.365	70	-1.674			150	-1.606	1770	-1.337
159	-1.370	75	-1.562	0.1	-0.474	200	-1.532	1780	-1.351
160	-1.374	80	-1.464	0.2	-0.985	250	-1.457	1790	-1.365
161	-1.378	85	-1.378	0.3	-1.378	300	-1.378	1800	-1.378
162	-1.383	90	-1.301	0.4	-1.710	350	-1.297	1810	-1.392
163	-1.387	95	-1.233	0.5	-2.003	400	-1.212	1820	-1.405
164	-1.391	100	-1.171	0.6	-2.268	450	-1.124	1830	-1.418
165	-1.395	105	-1.115	0.7	-2.512	500	-1.031	1840	-1.431
166	-1.399			0.8	-2.740	550	-0.933	1850	-1.444
						600	-0.828
Sensitivity = [(-1.383 + 1.374)/2]/ [1.378/161] = -0.53		Sensitivity = [(-1.301 + 1.464)/10]/ [1.378/85] = 1.00		Sensitivity = [(-1.710 + 0.985)/0.2]/ [1.378/0.3] = -0.79		Sensitivity = [(-1.297 + 1.457)/100]/ [1.378/300] = 0.35		Sensitivity = [(-1.392 + 1.365)/20]/ [1.378/1800] = -1.76
Range = 0.04		Range = 0.69		Range = 2.23		Range = 0.98		Range = 0.14

v_bulletafter	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch
		7	-1.359	20	-2.702	0.022	-1.356	0.005	-1.389
0	-1.254	8	-1.362	30	-2.530	0.023	-1.360	0.006	-1.387
50	-1.290	9	-1.366	40	-2.300	0.024	-1.365	0.007	-1.385
100	-1.322	10	-1.369	50	-2.022	0.025	-1.369	0.008	-1.383
150	-1.351	11	-1.374	60	-1.711	0.026	-1.374	0.009	-1.381
200	-1.378	12	-1.378	70	-1.378	0.027	-1.378	0.010	-1.378
250	-1.402	13	-1.383	80	-1.038	0.028	-1.383	0.011	-1.376
300	-1.422	14	-1.389	90	-0.705	0.029	-1.387	0.012	-1.374
350	-1.440	15	-1.394	100	-0.391	0.030	-1.391	0.013	-1.371
400	-1.454	16	-1.401	110	-0.106	0.031	-1.396	0.015	-1.369
		17	-1.407	120	+0.142	0.032	-1.400	0.015	-1.366
Sensitivity = [(-1.402 + 1.351)/100]/ [1.378/200] = -0.07		Sensitivity = [(-1.389 + 1.369)/4]/ [1.378/12] = -0.04		Sensitivity = [(-1.038 + 1.711)/20]/ [1.378/70] = 1.71		Sensitivity = [-1.387 + 1.369)/0.004]/ [1.378/0.027] = -0.09		Sensitivity = [(-1.374 + 1.383)/0.004]/ [1.378/0.010] = 0.02
Range = 0.20		Range = 0.05		Range = 2.84		Range = 0.04		Range = 0.02

v_frag1	v_lurch	v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_kefrag1	v_lurch
250	-1.366	250	-1.408					1.00	-1.420
300	-1.377	300	-1.406	20	-1.453	50	-1.426	1.05	-1.412
350	-1.384	350	-1.401	25	-1.439	55	-1.415	1.10	-1.403
400	-1.387	400	-1.395	30	-1.421	60	-1.403	1.15	-1.395
450	-1.385	450	-1.388	35	-1.401	65	-1.391	1.20	-1.387
500	-1.378	500	-1.378	40	-1.378	70	-1.378	1.25	-1.378
550	-1.367	550	-1.367	45	-1.353	75	-1.365	1.30	-1.370
600	-1.349	600	-1.355	50	-1.325	80	-1.351	1.35	-1.361
650	-1.327	650	-1.340	55	-1.295	85	-1.338	1.40	-1.353
700	-1.297	700	-1.324	60	-1.264	90	-1.324	1.45	-1.344
700	-1.261	750	-1.305					1.50	-1.335
Sensitivity = [(-1.367 + 1.385)/100]/ [1.378/500] = 0.06		Sensitivity = [(-1.367 + 1.388)/100]/ [1.378/500] = 0.08		Sensitivity = [(-1.353 + 1.401)/10]/ [1.378/40] = 0.14		Sensitivity = [(-1.365 + 1.391)/10]/ [1.378/70] = 0.13		Sensitivity = [(-1.370 + 1.387)/0.10]/ [1.378/1.25] = 0.15
Range = 0.10		Range = 0.10		Range = 0.19		Range = 0.10		Range = 0.08

f_kefrags23	v_lurch	f_I	v_lurch	f_kebody	v_lurch
1.00	-1.393	1.06	-1.443
1.05	-1.390	1.07	-1.429	1.00	-1.378
1.10	-1.387	1.08	-1.416	1.05	-1.378
1.15	-1.384	1.09	-1.403	1.10	-1.378
1.20	-1.381	1.10	-1.390	1.15	-1.378
1.25	-1.378	1.11	-1.378	1.20	-1.378
1.30	-1.375	1.12	-1.366	1.25	-1.378
1.35	-1.372	1.13	-1.354	1.30	-1.378
1.40	-1.369	1.14	-1.342	1.35	-1.378
1.45	-1.365	1.15	-1.330	1.40	-1.378
1.50	-1.362	1.16	-1.319
Sensitivity = [(-1.375 + 1.381)/0.10]/ [1.378/1.25] = 0.05		Sensitivity = [(-1.366 + 1.390)/0.02]/ [1.378/1.11] = 0.97		Sensitivity = [(-1.378 + 1.378)/0.40]/ [1.378/1.20] = 0.00
Range = 0.03		Range = 0.12		Range = 0.00

This ordered summary of sensitivities is virtually identical to the one for Lurch 4 Angular. Only a few terms have changed, and then only slightly. This is a consequence of the fact that the addition of 3-D movement of the body didn't change the solution perceptibly. As expected from that result, the new term, f_kebody, has an extremely small sensitivity and effect—both 0.00. As before, the two dominant sensitivities remained Q_cl and m_cloud, both properties of the cloud.

Summary
The addition of three-dimensionality to the motion of the body changed virtually nothing in the solution or the sensitivities. The variables of the cloud remained by far the most important. This means that the 3-D motion of the body is not a significant part of the solution.

m_bullet = 161 gr	v_bullet = 1800 ft s^-1	m_frag1 = 0.027 gr
m_body = 85 lb	v_bulletafter = 200 ft s^-1	m_frags23 = 0.01 gr
m_cloud = 0.3 lb	Q_frag1 = 40°	v_frag1 = 500 ft s^-1
Q = 12°	Q_frags23 = 70°	v_frags23 = 500 ft s^-1
PE = 300 ft-lb	Q_cl = 70°	f_kefrag1 = 1.25
f_I = 1.11	f_kebody = 1.2	f_kefrags23 = 1.25

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
f_kebody	0.00	0.00	Small
m_frags23	0.02	0.02	Small
f_kefrags23	0.05	0.03	Small
f_kefrag1	0.15	0.08	Small
v_frags23	0.08	0.10	Small
v_frag1	0.06	0.10	Small
Q_frags23	0.13	0.10	Small
f_I	0.97	0.12	Small
Q_frag1	0.14	0.19	Small
m_body	1.00	0.69	Medium
PE	0.35	0.98	Medium
Q_cl	1.71	2.84	Large
Negative effect on lurch (increases rearward velocity)
m_bullet	-0.53	0.04	Small
m_frag1	-0.09	0.04	Small
Q	-0.04	0.05	Small
v_bullet	-1.76	0.14	Small
v_bulletafter	-0.07	0.20	Small
m_cloud	-0.79	2.23	Large